# Seminars & Events for Department Colloquium

##### The sphere packing problem in dimensions 8 and 24.

The sphere packing problem is to find an arrangement of non-overlapping unit spheres in the $d$-dimensional Euclidean space in which the spheres fill as large a proportion of the space as possible. In this talk we will present a solution of the sphere packing problem in dimensions 8 and 24. In 2003 N. Elkies and H. Cohn proved that the existence of a real function satisfying certain constrains leads to an upper bound for the sphere packing constant. Using this method they obtained almost sharp estimates in dimensions 8 and 24. We will show that functions providing exact bounds can be constructed explicitly as certain integral transforms of modular forms. Therefore, the sphere packing problem in dimensions 8 and 24 is solved by a linear programming method.

##### Weyl law for the volume spectrum

We will present our proof that the volume spectrum of a Riemannian manifold satisfies a Weyl law. This was a conjecture of Gromov. The talk is based on joint work with Liokumovich and Neves.

##### Zeroes of harmonic functions and Laplace eigenfunctions: pursuing the conjectures by Yau and Nadirashvili

**Please note special date - THURSDAY, OCTOBER 13. **Nadirashvili conjectured that for any non-constant harmonic function in R^3 its zero set has infinite area. This question was motivated by the Yau conjecture on zero sets of Laplace eigenfunctions. Both conjectures can be treated as an attempt to control the zero set of a solution of elliptic PDE in terms of growth of the solution. For holomorhpic functions such kind of control is possible only from one side: there is a plenty of holomorphic functions that have no zeros. While for a real-valued harmonic function on a plane the length of the zero set can be estimated (locally) from above and below by the frequency, which is a characteristic of growth of the harmonic function.

##### A proof of Onsager's Conjecture

In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured that weak solutions to the incompressible Euler equations may violate the law of conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székel

##### The stability of Kerr-de Sitter black holes

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum. I will discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum.

##### Coincidences in homological densities

For certain natural sequences of topological spaces, the kth homology group stabilizes once you go far enough out in the sequence of spaces. This phenomenon is called homological stability. Two classical examples of homological stability are the configuration space of n unordered distinct points in the plane, studied in the 60's by Arnold' and the space of (based) algebraic maps from CP^1 to CP^1 studied by Segal in the 70's. It turns out that the stable homology is the same in these two examples, and in this talk we explain that this is just the tip an iceberg--a subtle, but precise relationship between the values of stable of homology different sequences of spaces. To explain this relationship, which we discovered through an analogy to asymptotic counts in number theory, we introduce a new notion of homological density.

##### TBD - Paul Seidel

##### Turbulent weak solutions of the Euler equations

**This joint Math/PACM colloquium will be held at 4:00, Monday, December 5, in Fine 214.**

Motivated by Kolmogorov's theory of hydrodynamic turbulence, we consider dissipative weak solutions to the 3D incompressible Euler equations. We show that there exist infinitely many weak solutions of the 3D Euler equations, which are continuous in time, lie in a Sobolev space $H^s$ with respect to space, and they do not conserve the kinetic energy. Here the smoothness parameter $s$ is at the Onsager critical value $1/3$, consistent with Kolmogorov's $-4/5$ law for the third order structure functions. We shall also discuss bounds for the second order structure functions, which deviate from the classical Kolmogorov 1941 theory. This talk is based on joint work with T. Buckmaster and N. Masmoudi.

##### Energy Identity for Stationary Yang Mills

The first part of this talk will introduce and discuss the basics of Yang Mills connections, which are connections over a principle bundle which are critical points of the energy functional \int |F|^2, the L^2 norm of the curvature of A, and thus A may be viewed as a solution to a nonlinear pde. In many problems, e.g. compactifications of moduli spaces, one considers sequences A_i of such connections which converge to a potentially singular limit connection A_i-> A . The convergence may not be smooth, and we can understand the blow up region by converging the energy measures |F_i|^2 dv_g -> |F|^2dv_g +\nu, where \nu=e(x)d\lambda^{n-4} is the n-4 rectifiable defect measure (e.g. think of \nu as being supported on an n-4 submanifold).

##### Subexponential growth, measure rigidity, strong property (T) and Zimmer's conjecture

Lattices in higher rank simple Lie groups, like SL(n,R) for n>2, are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds. After providing some history and motivation, I will discuss a very recent result, proving many cases of the main conjecture. While avoiding technical matters, I will try to describe some of the novel flavor of the proof.

##### Classical and quantum geometric Langlands via quantization in positive characteristic

Geometric Langlands is an algebro-geometric and categorified analog of the Langlands program, introduced by A.Beilinson and V.Drinfeld. It can be understood as a certain non-commutative Fourier-Mukai transform between two moduli spaces associated to an algebraic curve and a Langlands dual pair of algebraic groups. We will describe the approach to geometric Langlands via quantization in characteristic p, implemented by R.Bezrukavnikov and A.Braverman for the GL_N case (and generalized by T.-H. Chen and X.Zhu togeneral groups). In this context, due to a large center of algebras of differential operators in characteristic p, an (appropriately localized) version of the geometric Langlands equivalence can be established via an essentially commutative Fourier-Mukai transform for torsors over abelian schemes.

##### Metrics on surfaces which optimize eigenvalues

This will be a general lecture on the problem of finding metrics on surfaces with fixed area (or boundary length) which maximize eigenvalues. We will introduce the problem, discuss the geometry of optimal metrics, and describe general existence results.

##### Interpolation and Approximation

Suppose f is a real-valued function on an awful set E in R^n. How can we decide whether f extends to a smooth function F on the whole R^n? "Smooth" means that F belongs to our favorite space X of continuous functions, e.g. C^m, C^{m, alpha}, or W^{m,p}. If such an F exists, how small can we take its norm in X? What can we say about its derivatives at a given point in or near E? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F with close-to-minimal norm in X? How many computer operations does it take? What if we require merely that F agree approximately with f on E? What if we are allowed to discard a few points of E as "outliers"? Which points should we discard to make the norm of F as small as possible? The subject started with Whitney in 1934. I've been working on it for many years.

##### From the liquid drop model for nuclei to the ionization conjecture for atoms

The liquid drop model is an isoperimetric problem with a competing non-local term. It was originally introduced in the nuclear physics literature in 1930 and has received a lot of attention recently as an interesting problem in the calculus of variations. We discuss some new results and open problems. We show how the insights from this problem allowed us to prove the ionization conjecture in a certain model in density functional theory. The talk will be non-technical and does not assume any knowledge in physics.

##### Tableaux combinatorics of hopping particles and Koornwinder polynomials

The asymmetric simple exclusion process (ASEP) is a Markov chain describing particles hopping on a 1-dimensional finite lattice. Particles can enter and exit the lattice at the left and right boundaries, and particles can hop left and right in the lattice, subject to the condition that there is at most one particle per site. The ASEP has been cited as a model for traffic flow, protein synthesis, the nuclear pore complex, etc. In my talk I will discuss joint work with Corteel and with Corteel-Mandelshtam, in which we describe the stationary distribution of the ASEP and the 2-species ASEP using staircase tableaux and rhombic tilings. We also link these models to Askey Wilson polynomials and Macdonald-Koornwinder polynomials, which allows us to give combinatorial formulas for their moments.

##### The Waring problem and group theory

The Waring problem asks whether for each positive integer n, every positive integer is the sum of a bounded number of nth powers. Although Hilbert solved it more than 100 years ago, it remains the focus of much current research. I will discuss some questions in group theory, especially algebraic group theory, inspired by it. In particular, I will present work with Aner Shalev and Pham Tiep on Waring-type word maps on finite simple groups and work with Dong Quan Ngoc Nguyen on the unipotent Waring problem. The emphasis throughout will be on unsolved problems.

##### Statistics for random linear combinations of Laplace eigenfunctions

There are several questions about the behavior of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of their number of critical points, the study of the size of their zero set, the study of the number of connected components of their zero set, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for random linear combinations of eigenfunctions. In this talk I will present several results in this direction. This talk is based on joint works with Boris Hanin and Peter Sarnak.

##### A Variational approach to Liouville equations

Liouville equations arise when trying to uniformize curvatures or to extremise energies depending on spectra of surfaces. We consider cases when Gauss-Bonnet integrals are "large", as it might happen in presence of conical singularities. We prove existence of solutions combining geometric functional inequalities and a micro/macroscopic study of conformal volume distribution.

##### Ergodic Ramsey Theory at the Junction of Additive and Multiplicative Combinatorics

By analogy with the classical notions of density in the set N of natural numbers, one can introduce notions of density which are geared towards the multiplicative structure of N. Various combinatorial results involving additively large sets in (N,+) ( such as, for instance, Szemeredi's theorem on arithmetic progressions and its polynomial extensions) have natural analogs in the multiplicative semigroup (N,x). For example, multiplicatively large sets in N contain arbitrarily long geometric progressions. One can show, that, somewhat surprisingly, multiplicatively large sets contain also arbitrarily long arithmetic progressions. Some recent developments related to Sarnak's Mobius Disjointness Conjecture reveal new interesting connections between the theory of multiple recurrence and multiplicative number theory.